Rigging my human armature makes eyes be sucked into the skull
Just looking at the pictures, I am not positive but it could be that you need to fix the weight paint issues. Select the rig in Pose Mode and then Shift select the model (both should be selected) then go into Weight Paint (its in the Object/Edit mode menu. From there you can right click on whatever bone and see the weight influence. Red= full control and Blue is unaffected. If the eye are red then you can pull up the paint menu with T and select the Subtract brush. (Default is Draw). Weight is 1.0 meaning you will paint red and 0 is blue meaning you will kill the influence of the bone on the mesh. So if you lighten the color of the mesh affected by the eyebone then it should fix the problem.

If all that wasn't the issue, then try to delete doubles. Make sure your mesh has no double verts. Un-parent the mesh (Alt+P), delete all the vertex groups (go to the vertex tab it is next to the materials tab) and then select the mesh and press W and remove doubles. Then parent the mesh to the armature again. The order in case you messed it up is, select the mesh (child) then shift select the armature (parent) hit Ctrl+P parent with auto weights. Hopefully that fixes your issue. Please respond if it helped you fix your problem.